Uniform Circular Motion Lab

Uniform Circular Motion Lab: Exploring Centripetal Acceleration

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Circular motion is everywhere—from the rotation of car tires to satellites orbiting Earth to electrons circling atomic nuclei. What makes circular motion special is that even at constant speed, the object is constantly accelerating. This counterintuitive fact is the key to understanding centripetal force and the physics of curves, loops, and orbits.

This lab simulation lets you explore uniform circular motion by adjusting radius, period, and mass, while observing how these parameters affect velocity, centripetal acceleration, and centripetal force in real-time.

The Physics of Circular Motion

Why Acceleration Without Speeding Up?

An object moving in a circle at constant speed has constant speed but changing velocity. How is this possible? Remember that velocity is a vector—it has both magnitude (speed) and direction. In circular motion, the direction is constantly changing, even if the speed isn't [1].

This direction change requires acceleration, and that acceleration must point toward the center of the circle. We call it centripetal acceleration (from Latin centrum "center" + petere "to seek") [2].

Key Relationships

Essential Equations

From period to speed: $v = \frac{2\pi r}{T} = \omega r$

Centripetal acceleration: $a_c = \frac{v^2}{r} = \omega^2 r = \frac{4\pi^2 r}{T^2}$

Centripetal force: $F_c = ma_c = \frac{mv^2}{r} = m\omega^2 r$

Position as a function of time: $x(t) = r\cos(\omega t), \quad y(t) = r\sin(\omega t)$

Learning Objectives

After completing this lab, you should be able to:

1. Calculate period, frequency, and angular velocity from given data
2. Derive the relationship between linear speed and angular velocity
3. Explain why centripetal acceleration points toward the center
4. Predict how changing radius or period affects speed and acceleration
5. Apply the centripetal force equation to real-world scenarios
6. Interpret position vs. time graphs for circular motion (sinusoidal)

Exploration Activities

Activity 1: Speed-Radius Relationship

Objective: Verify that v = 2πr/T

Procedure:

1. Set T = 4 s (fixed period)
2. Record v for r = 1, 2, 3, 4 m
3. Plot v vs r—you should get a straight line through the origin
4. Calculate the slope and verify it equals 2π/T ≈ 1.57

Expected result: Speed is directly proportional to radius when period is constant.

Activity 2: Acceleration-Speed Relationship

Objective: Verify that a_c = v²/r

Procedure:

1. Keep r = 2 m constant
2. Vary period from T = 1 s to T = 4 s
3. Record both v and a_c for each trial
4. Plot a_c vs v²—should be a straight line
5. The slope should equal 1/r = 0.5

Key insight: Acceleration increases with the square of speed, making high-speed turns very demanding.

Activity 3: Force and Mass

Objective: Explore centripetal force requirements

Procedure:

1. Set r = 2 m and T = 2 s
2. Vary mass from 0.5 kg to 5 kg
3. Record F_c for each mass
4. Plot F_c vs m—straight line through origin
5. Verify the slope equals v²/r

Discussion: Why does a heavier car need better brakes for the same turn?

Activity 4: Period and Frequency

Objective: Understand the inverse relationship

Procedure:

1. Use "Set Frequency" mode
2. Set f = 0.5 Hz and record T
3. Double f to 1.0 Hz and record T
4. Verify T × f = 1 for all cases

Real-World Applications

Reference Data

Data from NASA and standard physics references [3].

Challenge Questions

Level 1 (Basic):

1. An object completes 5 revolutions in 20 seconds. What is its period? What is its frequency?

Level 2 (Intermediate): 2. A 2 kg mass moves in a circle of radius 0.5 m at 4 m/s. Calculate the centripetal force.

Level 3 (Advanced): 3. A car travels around a flat circular track at 25 m/s. If the coefficient of static friction is 0.8, what is the minimum radius of the track?

Level 4 (Challenge): 4. Derive the formula a_c = v²/r using calculus (differentiate the position vector).

Level 5 (Extension): 5. A ball on a string is swung in a vertical circle. At what point is the tension in the string maximum? At what speed at the top will the string go slack?

Common Mistakes to Avoid

Frequently Asked Questions

Q: Why doesn't the object fly outward if force points inward? A: The inward force doesn't pull the object to the center—it constantly deflects the object from its straight-line path. Without this force, the object would fly off tangentially (Newton's First Law), not radially [1].

Q: What provides the centripetal force in different situations? A: Different forces can act as centripetal force: tension (ball on string), friction (car turning), gravity (satellites), normal force (banked curves), or electromagnetic force (particles in accelerators) [2].

Q: Why does the formula have v² and not just v? A: The v² arises from the geometry of circular motion. Both the magnitude of velocity change (proportional to v) and the rate of that change (also proportional to v) contribute, giving v² overall [1].

Q: How fast would I need to swing a bucket of water overhead without spilling? A: At the top of the circle, you need mg ≤ mv²/r, so v ≥ √(gr). For r = 1 m and g = 9.81 m/s², you need v ≥ 3.13 m/s [4].

Q: Why do I feel pushed outward on a carousel? A: In your rotating reference frame, you experience a fictitious centrifugal force pushing you outward. In an inertial frame, you're simply trying to go straight while the carousel floor pushes you inward [2].

References

[1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics, 10th Edition. Wiley. Chapter 4: Motion in Two and Three Dimensions.

[2] Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers, 10th Edition. Cengage Learning. Chapter 6: Circular Motion and Other Applications of Newton's Laws.

[3] NASA. (2024). Planetary Fact Sheets. NASA Goddard Space Flight Center.

[4] Knight, R. D. (2016). Physics for Scientists and Engineers: A Strategic Approach, 4th Edition. Pearson.

About the Data

Orbital data sourced from NASA planetary fact sheets. The physics equations follow standard notation from university physics textbooks. All calculations in the simulation use SI units.

Citation Guide

To cite this simulation in academic work:

Simulations4All. (2025). Uniform Circular Motion Lab [Interactive simulation]. Simulations4All Educational Platform. Retrieved from https://simulations4all.com/simulations/ucm-formulas-lab

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